The Brouwer-Kakutani Fixed Point Theorem


	The Brouwer-Kakutani Fixed Point Theorem

wildcat

1-23-2008 10:36 AM

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tags:

math, evolution, mapping

wildcat says:

fascinating read

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<div style="margin: 12px 0px; font-family: arial; color: #333333; background: #ffffff; border: solid 4px #e5e5e5; width: 100%; clear: left;"><div class="CM_CTB_Content_Wrap" style="margin: 0px; padding: 0px;background-color: #ffffff;"><div style="border-bottom: solid 1px #dcdcdc; white-space: nowrap; margin-bottom: 8px; background-color: #eeeeee ;background-image: url(http://www.clipmarks.com/images/source-bg.gif); background-repeat: repeat-x; height: 24px; line-height: 24px; vertical-align: middle; padding-bottom: 4px; color: #666666; font-size: 10px;" ><a href="http://clipmarks.com/clip-to-blog/" title="see clips that are hot right now"><img src="http://content.clipmarks.com/blog_embed/ad30a07d-19b7-4998-960b-70d4ecff5243/4173544A-C180-404A-8319-A29BB46A8F7D/" alt="" width="19" height="19" border="0" style="vertical-align: middle; margin: 0px 4px; display: inline; border: none; float:none;" /></a>clipped from <a title="http://underground.musenet.org:8080/utnebury/fixed.point.html" href="http://underground.musenet.org:8080/utnebury/fixed.point.html" style="font-size: 11px;">underground.musenet.org:8080</a></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://underground.musenet.org:8080/utnebury/fixed.point.html">
There is a famous Theorem in modern mathematics, called
the Fixed Point Theorem, attributed
to L. Brouwer, and later clarified by Kakutani.
To explain the Theorem in simple terms,
consider a mathematical function
which maps each point in a given space onto a
corresponding point in the same space.
A popular version of such a map is an ordinary hand calculator.
Any given calculation (i.e. function) transforms the number
shown in the display into another number shown in the display.
For the calculator example, the space is just the set of possible
numbers that can appear in the display.
The Fixed Point Theorem states the conditions under which
a function possesses a point that maps into itself.
<P>
</P><H3></H3></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://underground.musenet.org:8080/utnebury/fixed.point.html">One can hypothesize a possible future state in which all elements
of the culture adopt common goals of mutual survival, mutual well-being,
mutual growth, and mutual fulfillment</blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://underground.musenet.org:8080/utnebury/fixed.point.html">It appears that we are very far from adopting such goals</blockquote></div><div style="margin: 0px 6px 6px 4px;"><table style="font-size: 11px;border-spacing: 0px;padding: 0px;" cellpadding="0" cellspacing="0" width="100%"><tr><td style="background:transparent;border-width:0px;padding:0px;">&nbsp;</td><td align="right" style="background:transparent;border-width:0px;padding:0px;width:107px" width="107"><a href="http://clipmarks.com/share/4173544A-C180-404A-8319-A29BB46A8F7D/blog/" title="blog or email this clip"><img src="http://content8.clipmarks.com/images/c2b-foot.png" border="0" alt="blog it" width="107" height="17" style="border-width:0px;padding:0px;margin:0px;" /></a></td></tr></table></div></div>

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